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authorSon Ho2023-06-09 10:06:43 +0200
committerSon Ho2023-06-09 10:06:43 +0200
commite2fef1a5c986aff4c9975b1376bcc0fc0bb87940 (patch)
treed1bd1ee2170544d9ad878e708ac3c46558851dcc /backends/lean/Base
parentacc09d5c69690f2c46cb1bacf290da5dcc268b24 (diff)
Reorganize a bit the Lean library
Diffstat (limited to 'backends/lean/Base')
-rw-r--r--backends/lean/Base/Primitives.lean650
1 files changed, 650 insertions, 0 deletions
diff --git a/backends/lean/Base/Primitives.lean b/backends/lean/Base/Primitives.lean
new file mode 100644
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--- /dev/null
+++ b/backends/lean/Base/Primitives.lean
@@ -0,0 +1,650 @@
+import Lean
+import Lean.Meta.Tactic.Simp
+import Init.Data.List.Basic
+import Mathlib.Tactic.RunCmd
+import Mathlib.Tactic.Linarith
+
+--------------------
+-- ASSERT COMMAND --Std.
+--------------------
+
+open Lean Elab Command Term Meta
+
+syntax (name := assert) "#assert" term: command
+
+@[command_elab assert]
+unsafe
+def assertImpl : CommandElab := fun (_stx: Syntax) => do
+ runTermElabM (fun _ => do
+ let r ← evalTerm Bool (mkConst ``Bool) _stx[1]
+ if not r then
+ logInfo "Assertion failed for: "
+ logInfo _stx[1]
+ logError "Expression reduced to false"
+ pure ())
+
+#eval 2 == 2
+#assert (2 == 2)
+
+-------------
+-- PRELUDE --
+-------------
+
+-- Results & monadic combinators
+
+inductive Error where
+ | assertionFailure: Error
+ | integerOverflow: Error
+ | divisionByZero: Error
+ | arrayOutOfBounds: Error
+ | maximumSizeExceeded: Error
+ | panic: Error
+deriving Repr, BEq
+
+open Error
+
+inductive Result (α : Type u) where
+ | ret (v: α): Result α
+ | fail (e: Error): Result α
+deriving Repr, BEq
+
+open Result
+
+instance Result_Inhabited (α : Type u) : Inhabited (Result α) :=
+ Inhabited.mk (fail panic)
+
+/- HELPERS -/
+
+def ret? {α: Type} (r: Result α): Bool :=
+ match r with
+ | Result.ret _ => true
+ | Result.fail _ => false
+
+def massert (b:Bool) : Result Unit :=
+ if b then .ret () else fail assertionFailure
+
+def eval_global {α: Type} (x: Result α) (_: ret? x): α :=
+ match x with
+ | Result.fail _ => by contradiction
+ | Result.ret x => x
+
+/- DO-DSL SUPPORT -/
+
+def bind (x: Result α) (f: α -> Result β) : Result β :=
+ match x with
+ | ret v => f v
+ | fail v => fail v
+
+-- Allows using Result in do-blocks
+instance : Bind Result where
+ bind := bind
+
+-- Allows using return x in do-blocks
+instance : Pure Result where
+ pure := fun x => ret x
+
+/- CUSTOM-DSL SUPPORT -/
+
+-- Let-binding the Result of a monadic operation is oftentimes not sufficient,
+-- because we may need a hypothesis for equational reasoning in the scope. We
+-- rely on subtype, and a custom let-binding operator, in effect recreating our
+-- own variant of the do-dsl
+
+def Result.attach {α: Type} (o : Result α): Result { x : α // o = ret x } :=
+ match o with
+ | .ret x => .ret ⟨x, rfl⟩
+ | .fail e => .fail e
+
+macro "let" e:term " ⟵ " f:term : doElem =>
+ `(doElem| let ⟨$e, h⟩ ← Result.attach $f)
+
+-- TODO: any way to factorize both definitions?
+macro "let" e:term " <-- " f:term : doElem =>
+ `(doElem| let ⟨$e, h⟩ ← Result.attach $f)
+
+-- We call the hypothesis `h`, in effect making it unavailable to the user
+-- (because too much shadowing). But in practice, once can use the French single
+-- quote notation (input with f< and f>), where `‹ h ›` finds a suitable
+-- hypothesis in the context, this is equivalent to `have x: h := by assumption in x`
+#eval do
+ let y <-- .ret (0: Nat)
+ let _: y = 0 := by cases ‹ ret 0 = ret y › ; decide
+ let r: { x: Nat // x = 0 } := ⟨ y, by assumption ⟩
+ .ret r
+
+----------------------
+-- MACHINE INTEGERS --
+----------------------
+
+-- We redefine our machine integers types.
+
+-- For Isize/Usize, we reuse `getNumBits` from `USize`. You cannot reduce `getNumBits`
+-- using the simplifier, meaning that proofs do not depend on the compile-time value of
+-- USize.size. (Lean assumes 32 or 64-bit platforms, and Rust doesn't really support, at
+-- least officially, 16-bit microcontrollers, so this seems like a fine design decision
+-- for now.)
+
+-- Note from Chris Bailey: "If there's more than one salient property of your
+-- definition then the subtyping strategy might get messy, and the property part
+-- of a subtype is less discoverable by the simplifier or tactics like
+-- library_search." So, we will not add refinements on the return values of the
+-- operations defined on Primitives, but will rather rely on custom lemmas to
+-- invert on possible return values of the primitive operations.
+
+-- Machine integer constants, done via `ofNatCore`, which requires a proof that
+-- the `Nat` fits within the desired integer type. We provide a custom tactic.
+
+open System.Platform.getNumBits
+
+-- TODO: is there a way of only importing System.Platform.getNumBits?
+--
+@[simp] def size_num_bits : Nat := (System.Platform.getNumBits ()).val
+
+-- Remark: Lean seems to use < for the comparisons with the upper bounds by convention.
+-- We keep the F* convention for now.
+@[simp] def Isize.min : Int := - (HPow.hPow 2 (size_num_bits - 1))
+@[simp] def Isize.max : Int := (HPow.hPow 2 (size_num_bits - 1)) - 1
+@[simp] def I8.min : Int := - (HPow.hPow 2 7)
+@[simp] def I8.max : Int := HPow.hPow 2 7 - 1
+@[simp] def I16.min : Int := - (HPow.hPow 2 15)
+@[simp] def I16.max : Int := HPow.hPow 2 15 - 1
+@[simp] def I32.min : Int := -(HPow.hPow 2 31)
+@[simp] def I32.max : Int := HPow.hPow 2 31 - 1
+@[simp] def I64.min : Int := -(HPow.hPow 2 63)
+@[simp] def I64.max : Int := HPow.hPow 2 63 - 1
+@[simp] def I128.min : Int := -(HPow.hPow 2 127)
+@[simp] def I128.max : Int := HPow.hPow 2 127 - 1
+@[simp] def Usize.min : Int := 0
+@[simp] def Usize.max : Int := HPow.hPow 2 size_num_bits - 1
+@[simp] def U8.min : Int := 0
+@[simp] def U8.max : Int := HPow.hPow 2 8 - 1
+@[simp] def U16.min : Int := 0
+@[simp] def U16.max : Int := HPow.hPow 2 16 - 1
+@[simp] def U32.min : Int := 0
+@[simp] def U32.max : Int := HPow.hPow 2 32 - 1
+@[simp] def U64.min : Int := 0
+@[simp] def U64.max : Int := HPow.hPow 2 64 - 1
+@[simp] def U128.min : Int := 0
+@[simp] def U128.max : Int := HPow.hPow 2 128 - 1
+
+#assert (I8.min == -128)
+#assert (I8.max == 127)
+#assert (I16.min == -32768)
+#assert (I16.max == 32767)
+#assert (I32.min == -2147483648)
+#assert (I32.max == 2147483647)
+#assert (I64.min == -9223372036854775808)
+#assert (I64.max == 9223372036854775807)
+#assert (I128.min == -170141183460469231731687303715884105728)
+#assert (I128.max == 170141183460469231731687303715884105727)
+#assert (U8.min == 0)
+#assert (U8.max == 255)
+#assert (U16.min == 0)
+#assert (U16.max == 65535)
+#assert (U32.min == 0)
+#assert (U32.max == 4294967295)
+#assert (U64.min == 0)
+#assert (U64.max == 18446744073709551615)
+#assert (U128.min == 0)
+#assert (U128.max == 340282366920938463463374607431768211455)
+
+inductive ScalarTy :=
+| Isize
+| I8
+| I16
+| I32
+| I64
+| I128
+| Usize
+| U8
+| U16
+| U32
+| U64
+| U128
+
+def Scalar.min (ty : ScalarTy) : Int :=
+ match ty with
+ | .Isize => Isize.min
+ | .I8 => I8.min
+ | .I16 => I16.min
+ | .I32 => I32.min
+ | .I64 => I64.min
+ | .I128 => I128.min
+ | .Usize => Usize.min
+ | .U8 => U8.min
+ | .U16 => U16.min
+ | .U32 => U32.min
+ | .U64 => U64.min
+ | .U128 => U128.min
+
+def Scalar.max (ty : ScalarTy) : Int :=
+ match ty with
+ | .Isize => Isize.max
+ | .I8 => I8.max
+ | .I16 => I16.max
+ | .I32 => I32.max
+ | .I64 => I64.max
+ | .I128 => I128.max
+ | .Usize => Usize.max
+ | .U8 => U8.max
+ | .U16 => U16.max
+ | .U32 => U32.max
+ | .U64 => U64.max
+ | .U128 => U128.max
+
+-- "Conservative" bounds
+-- We use those because we can't compare to the isize bounds (which can't
+-- reduce at compile-time). Whenever we perform an arithmetic operation like
+-- addition we need to check that the result is in bounds: we first compare
+-- to the conservative bounds, which reduce, then compare to the real bounds.
+-- This is useful for the various #asserts that we want to reduce at
+-- type-checking time.
+def Scalar.cMin (ty : ScalarTy) : Int :=
+ match ty with
+ | .Isize => I32.min
+ | _ => Scalar.min ty
+
+def Scalar.cMax (ty : ScalarTy) : Int :=
+ match ty with
+ | .Isize => I32.max
+ | .Usize => U32.max
+ | _ => Scalar.max ty
+
+theorem Scalar.cMin_bound ty : Scalar.min ty ≤ Scalar.cMin ty := by
+ cases ty <;> simp [Scalar.min, Scalar.max, Scalar.cMin, Scalar.cMax] at *
+ cases System.Platform.numBits_eq <;>
+ unfold System.Platform.numBits at * <;>
+ simp [*]
+
+theorem Scalar.cMax_bound ty : Scalar.cMax ty ≤ Scalar.max ty := by
+ cases ty <;> simp [Scalar.min, Scalar.max, Scalar.cMin, Scalar.cMax] at * <;>
+ cases System.Platform.numBits_eq <;>
+ unfold System.Platform.numBits at * <;>
+ simp [*]
+
+theorem Scalar.cMin_suffices ty (h : Scalar.cMin ty ≤ x) : Scalar.min ty ≤ x := by
+ cases ty <;> simp [Scalar.min, Scalar.max, Scalar.cMin, Scalar.cMax] at * <;>
+ cases System.Platform.numBits_eq <;>
+ unfold System.Platform.numBits at * <;>
+ simp [*] at *
+ -- TODO: I would have expected terms like `-(1 + 1) ^ 63` to be simplified
+ linarith
+
+theorem Scalar.cMax_suffices ty (h : x ≤ Scalar.cMax ty) : x ≤ Scalar.max ty := by
+ cases ty <;> simp [Scalar.min, Scalar.max, Scalar.cMin, Scalar.cMax] at * <;>
+ cases System.Platform.numBits_eq <;>
+ unfold System.Platform.numBits at * <;>
+ simp [*] at * <;>
+ -- TODO: I would have expected terms like `-(1 + 1) ^ 63` to be simplified
+ linarith
+
+structure Scalar (ty : ScalarTy) where
+ val : Int
+ hmin : Scalar.min ty ≤ val
+ hmax : val ≤ Scalar.max ty
+deriving Repr
+
+theorem Scalar.bound_suffices (ty : ScalarTy) (x : Int) :
+ Scalar.cMin ty ≤ x ∧ x ≤ Scalar.cMax ty ->
+ Scalar.min ty ≤ x ∧ x ≤ Scalar.max ty
+ :=
+ λ h => by
+ apply And.intro <;> have hmin := Scalar.cMin_bound ty <;> have hmax := Scalar.cMax_bound ty <;> linarith
+
+def Scalar.ofIntCore {ty : ScalarTy} (x : Int)
+ (hmin : Scalar.min ty ≤ x) (hmax : x ≤ Scalar.max ty) : Scalar ty :=
+ { val := x, hmin := hmin, hmax := hmax }
+
+def Scalar.ofInt {ty : ScalarTy} (x : Int)
+ (h : Scalar.min ty ≤ x ∧ x ≤ Scalar.max ty) : Scalar ty :=
+ -- Remark: we initially wrote:
+ -- let ⟨ hmin, hmax ⟩ := h
+ -- Scalar.ofIntCore x hmin hmax
+ -- We updated to the line below because a similar pattern in `Scalar.tryMk`
+ -- made reduction block. Both versions seem to work for `Scalar.ofInt`, though.
+ -- TODO: investigate
+ Scalar.ofIntCore x h.left h.right
+
+@[simp] def Scalar.check_bounds (ty : ScalarTy) (x : Int) : Bool :=
+ (Scalar.cMin ty ≤ x || Scalar.min ty ≤ x) ∧ (x ≤ Scalar.cMax ty || x ≤ Scalar.max ty)
+
+theorem Scalar.check_bounds_prop {ty : ScalarTy} {x : Int} (h: Scalar.check_bounds ty x) :
+ Scalar.min ty ≤ x ∧ x ≤ Scalar.max ty := by
+ simp at *
+ have ⟨ hmin, hmax ⟩ := h
+ have hbmin := Scalar.cMin_bound ty
+ have hbmax := Scalar.cMax_bound ty
+ cases hmin <;> cases hmax <;> apply And.intro <;> linarith
+
+-- Further thoughts: look at what has been done here:
+-- https://github.com/leanprover-community/mathlib4/blob/master/Mathlib/Data/Fin/Basic.lean
+-- and
+-- https://github.com/leanprover-community/mathlib4/blob/master/Mathlib/Data/UInt.lean
+-- which both contain a fair amount of reasoning already!
+def Scalar.tryMk (ty : ScalarTy) (x : Int) : Result (Scalar ty) :=
+ if h:Scalar.check_bounds ty x then
+ -- If we do:
+ -- ```
+ -- let ⟨ hmin, hmax ⟩ := (Scalar.check_bounds_prop h)
+ -- Scalar.ofIntCore x hmin hmax
+ -- ```
+ -- then normalization blocks (for instance, some proofs which use reflexivity fail).
+ -- However, the version below doesn't block reduction (TODO: investigate):
+ return Scalar.ofInt x (Scalar.check_bounds_prop h)
+ else fail integerOverflow
+
+def Scalar.neg {ty : ScalarTy} (x : Scalar ty) : Result (Scalar ty) := Scalar.tryMk ty (- x.val)
+
+def Scalar.div {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) :=
+ if y.val != 0 then Scalar.tryMk ty (x.val / y.val) else fail divisionByZero
+
+-- Our custom remainder operation, which satisfies the semantics of Rust
+-- TODO: is there a better way?
+def scalar_rem (x y : Int) : Int :=
+ if 0 ≤ x then |x| % |y|
+ else - (|x| % |y|)
+
+-- Our custom division operation, which satisfies the semantics of Rust
+-- TODO: is there a better way?
+def scalar_div (x y : Int) : Int :=
+ if 0 ≤ x && 0 ≤ y then |x| / |y|
+ else if 0 ≤ x && y < 0 then - (|x| / |y|)
+ else if x < 0 && 0 ≤ y then - (|x| / |y|)
+ else |x| / |y|
+
+-- Checking that the remainder operation is correct
+#assert scalar_rem 1 2 = 1
+#assert scalar_rem (-1) 2 = -1
+#assert scalar_rem 1 (-2) = 1
+#assert scalar_rem (-1) (-2) = -1
+#assert scalar_rem 7 3 = (1:Int)
+#assert scalar_rem (-7) 3 = -1
+#assert scalar_rem 7 (-3) = 1
+#assert scalar_rem (-7) (-3) = -1
+
+-- Checking that the division operation is correct
+#assert scalar_div 3 2 = 1
+#assert scalar_div (-3) 2 = -1
+#assert scalar_div 3 (-2) = -1
+#assert scalar_div (-3) (-2) = 1
+#assert scalar_div 7 3 = 2
+#assert scalar_div (-7) 3 = -2
+#assert scalar_div 7 (-3) = -2
+#assert scalar_div (-7) (-3) = 2
+
+def Scalar.rem {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) :=
+ if y.val != 0 then Scalar.tryMk ty (x.val % y.val) else fail divisionByZero
+
+def Scalar.add {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) :=
+ Scalar.tryMk ty (x.val + y.val)
+
+def Scalar.sub {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) :=
+ Scalar.tryMk ty (x.val - y.val)
+
+def Scalar.mul {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) :=
+ Scalar.tryMk ty (x.val * y.val)
+
+-- TODO: instances of +, -, * etc. for scalars
+
+-- Cast an integer from a [src_ty] to a [tgt_ty]
+-- TODO: check the semantics of casts in Rust
+def Scalar.cast {src_ty : ScalarTy} (tgt_ty : ScalarTy) (x : Scalar src_ty) : Result (Scalar tgt_ty) :=
+ Scalar.tryMk tgt_ty x.val
+
+-- The scalar types
+-- We declare the definitions as reducible so that Lean can unfold them (useful
+-- for type class resolution for instance).
+@[reducible] def Isize := Scalar .Isize
+@[reducible] def I8 := Scalar .I8
+@[reducible] def I16 := Scalar .I16
+@[reducible] def I32 := Scalar .I32
+@[reducible] def I64 := Scalar .I64
+@[reducible] def I128 := Scalar .I128
+@[reducible] def Usize := Scalar .Usize
+@[reducible] def U8 := Scalar .U8
+@[reducible] def U16 := Scalar .U16
+@[reducible] def U32 := Scalar .U32
+@[reducible] def U64 := Scalar .U64
+@[reducible] def U128 := Scalar .U128
+
+-- TODO: below: not sure this is the best way.
+-- Should we rather overload operations like +, -, etc.?
+-- Also, it is possible to automate the generation of those definitions
+-- with macros (but would it be a good idea? It would be less easy to
+-- read the file, which is not supposed to change a lot)
+
+-- Negation
+
+/--
+Remark: there is no heterogeneous negation in the Lean prelude: we thus introduce
+one here.
+
+The notation typeclass for heterogeneous addition.
+This enables the notation `- a : β` where `a : α`.
+-/
+class HNeg (α : Type u) (β : outParam (Type v)) where
+ /-- `- a` computes the negation of `a`.
+ The meaning of this notation is type-dependent. -/
+ hNeg : α → β
+
+prefix:75 "-" => HNeg.hNeg
+
+instance : HNeg Isize (Result Isize) where hNeg x := Scalar.neg x
+instance : HNeg I8 (Result I8) where hNeg x := Scalar.neg x
+instance : HNeg I16 (Result I16) where hNeg x := Scalar.neg x
+instance : HNeg I32 (Result I32) where hNeg x := Scalar.neg x
+instance : HNeg I64 (Result I64) where hNeg x := Scalar.neg x
+instance : HNeg I128 (Result I128) where hNeg x := Scalar.neg x
+
+-- Addition
+instance {ty} : HAdd (Scalar ty) (Scalar ty) (Result (Scalar ty)) where
+ hAdd x y := Scalar.add x y
+
+-- Substraction
+instance {ty} : HSub (Scalar ty) (Scalar ty) (Result (Scalar ty)) where
+ hSub x y := Scalar.sub x y
+
+-- Multiplication
+instance {ty} : HMul (Scalar ty) (Scalar ty) (Result (Scalar ty)) where
+ hMul x y := Scalar.mul x y
+
+-- Division
+instance {ty} : HDiv (Scalar ty) (Scalar ty) (Result (Scalar ty)) where
+ hDiv x y := Scalar.div x y
+
+-- Remainder
+instance {ty} : HMod (Scalar ty) (Scalar ty) (Result (Scalar ty)) where
+ hMod x y := Scalar.rem x y
+
+-- ofIntCore
+-- TODO: typeclass?
+def Isize.ofIntCore := @Scalar.ofIntCore .Isize
+def I8.ofIntCore := @Scalar.ofIntCore .I8
+def I16.ofIntCore := @Scalar.ofIntCore .I16
+def I32.ofIntCore := @Scalar.ofIntCore .I32
+def I64.ofIntCore := @Scalar.ofIntCore .I64
+def I128.ofIntCore := @Scalar.ofIntCore .I128
+def Usize.ofIntCore := @Scalar.ofIntCore .Usize
+def U8.ofIntCore := @Scalar.ofIntCore .U8
+def U16.ofIntCore := @Scalar.ofIntCore .U16
+def U32.ofIntCore := @Scalar.ofIntCore .U32
+def U64.ofIntCore := @Scalar.ofIntCore .U64
+def U128.ofIntCore := @Scalar.ofIntCore .U128
+
+-- ofInt
+-- TODO: typeclass?
+def Isize.ofInt := @Scalar.ofInt .Isize
+def I8.ofInt := @Scalar.ofInt .I8
+def I16.ofInt := @Scalar.ofInt .I16
+def I32.ofInt := @Scalar.ofInt .I32
+def I64.ofInt := @Scalar.ofInt .I64
+def I128.ofInt := @Scalar.ofInt .I128
+def Usize.ofInt := @Scalar.ofInt .Usize
+def U8.ofInt := @Scalar.ofInt .U8
+def U16.ofInt := @Scalar.ofInt .U16
+def U32.ofInt := @Scalar.ofInt .U32
+def U64.ofInt := @Scalar.ofInt .U64
+def U128.ofInt := @Scalar.ofInt .U128
+
+-- Comparisons
+instance {ty} : LT (Scalar ty) where
+ lt a b := LT.lt a.val b.val
+
+instance {ty} : LE (Scalar ty) where le a b := LE.le a.val b.val
+
+instance Scalar.decLt {ty} (a b : Scalar ty) : Decidable (LT.lt a b) := Int.decLt ..
+instance Scalar.decLe {ty} (a b : Scalar ty) : Decidable (LE.le a b) := Int.decLe ..
+
+theorem Scalar.eq_of_val_eq {ty} : ∀ {i j : Scalar ty}, Eq i.val j.val → Eq i j
+ | ⟨_, _, _⟩, ⟨_, _, _⟩, rfl => rfl
+
+theorem Scalar.val_eq_of_eq {ty} {i j : Scalar ty} (h : Eq i j) : Eq i.val j.val :=
+ h ▸ rfl
+
+theorem Scalar.ne_of_val_ne {ty} {i j : Scalar ty} (h : Not (Eq i.val j.val)) : Not (Eq i j) :=
+ fun h' => absurd (val_eq_of_eq h') h
+
+instance (ty : ScalarTy) : DecidableEq (Scalar ty) :=
+ fun i j =>
+ match decEq i.val j.val with
+ | isTrue h => isTrue (Scalar.eq_of_val_eq h)
+ | isFalse h => isFalse (Scalar.ne_of_val_ne h)
+
+def Scalar.toInt {ty} (n : Scalar ty) : Int := n.val
+
+-- Tactic to prove that integers are in bounds
+syntax "intlit" : tactic
+
+macro_rules
+ | `(tactic| intlit) => `(tactic| apply Scalar.bound_suffices ; decide)
+
+-- -- We now define a type class that subsumes the various machine integer types, so
+-- -- as to write a concise definition for scalar_cast, rather than exhaustively
+-- -- enumerating all of the possible pairs. We remark that Rust has sane semantics
+-- -- and fails if a cast operation would involve a truncation or modulo.
+
+-- class MachineInteger (t: Type) where
+-- size: Nat
+-- val: t -> Fin size
+-- ofNatCore: (n:Nat) -> LT.lt n size -> t
+
+-- set_option hygiene false in
+-- run_cmd
+-- for typeName in [`UInt8, `UInt16, `UInt32, `UInt64, `USize].map Lean.mkIdent do
+-- Lean.Elab.Command.elabCommand (← `(
+-- namespace $typeName
+-- instance: MachineInteger $typeName where
+-- size := size
+-- val := val
+-- ofNatCore := ofNatCore
+-- end $typeName
+-- ))
+
+-- -- Aeneas only instantiates the destination type (`src` is implicit). We rely on
+-- -- Lean to infer `src`.
+
+-- def scalar_cast { src: Type } (dst: Type) [ MachineInteger src ] [ MachineInteger dst ] (x: src): Result dst :=
+-- if h: MachineInteger.val x < MachineInteger.size dst then
+-- .ret (MachineInteger.ofNatCore (MachineInteger.val x).val h)
+-- else
+-- .fail integerOverflow
+
+-------------
+-- VECTORS --
+-------------
+
+def Vec (α : Type u) := { l : List α // List.length l ≤ Usize.max }
+
+def vec_new (α : Type u): Vec α := ⟨ [], by apply Scalar.cMax_suffices .Usize; simp ⟩
+
+def vec_len (α : Type u) (v : Vec α) : Usize :=
+ let ⟨ v, l ⟩ := v
+ Usize.ofIntCore (List.length v) (by simp [Scalar.min]) l
+
+def vec_push_fwd (α : Type u) (_ : Vec α) (_ : α) : Unit := ()
+
+def vec_push_back (α : Type u) (v : Vec α) (x : α) : Result (Vec α)
+ :=
+ let nlen := List.length v.val + 1
+ if h : nlen ≤ U32.max || nlen ≤ Usize.max then
+ have h : nlen ≤ Usize.max := by
+ simp at *
+ cases System.Platform.numBits_eq <;>
+ unfold System.Platform.numBits at * <;>
+ simp [*] at * <;>
+ try assumption
+ cases h <;>
+ linarith
+ return ⟨ List.concat v.val x, by simp at *; assumption ⟩
+ else
+ fail maximumSizeExceeded
+
+def vec_insert_fwd (α : Type u) (v: Vec α) (i: Usize) (_: α): Result Unit :=
+ if i.val < List.length v.val then
+ .ret ()
+ else
+ .fail arrayOutOfBounds
+
+def vec_insert_back (α : Type u) (v: Vec α) (i: Usize) (x: α): Result (Vec α) :=
+ if i.val < List.length v.val then
+ -- TODO: maybe we should redefine a list library which uses integers
+ -- (instead of natural numbers)
+ let i := i.val.toNat
+ .ret ⟨ List.set v.val i x, by
+ have h: List.length v.val ≤ Usize.max := v.property
+ simp [*] at *
+ assumption
+ ⟩
+ else
+ .fail arrayOutOfBounds
+
+def vec_index_to_fin {α : Type u} {v: Vec α} {i: Usize} (h : i.val < List.length v.val) :
+ Fin (List.length v.val) :=
+ let j := i.val.toNat
+ let h: j < List.length v.val := by
+ have heq := @Int.toNat_lt (List.length v.val) i.val i.hmin
+ apply heq.mpr
+ assumption
+ ⟨j, h⟩
+
+def vec_index_fwd (α : Type u) (v: Vec α) (i: Usize): Result α :=
+ if h: i.val < List.length v.val then
+ let i := vec_index_to_fin h
+ .ret (List.get v.val i)
+ else
+ .fail arrayOutOfBounds
+
+def vec_index_back (α : Type u) (v: Vec α) (i: Usize) (_: α): Result Unit :=
+ if i.val < List.length v.val then
+ .ret ()
+ else
+ .fail arrayOutOfBounds
+
+def vec_index_mut_fwd (α : Type u) (v: Vec α) (i: Usize): Result α :=
+ if h: i.val < List.length v.val then
+ let i := vec_index_to_fin h
+ .ret (List.get v.val i)
+ else
+ .fail arrayOutOfBounds
+
+def vec_index_mut_back (α : Type u) (v: Vec α) (i: Usize) (x: α): Result (Vec α) :=
+ if h: i.val < List.length v.val then
+ let i := vec_index_to_fin h
+ .ret ⟨ List.set v.val i x, by
+ have h: List.length v.val ≤ Usize.max := v.property
+ simp [*] at *
+ assumption
+ ⟩
+ else
+ .fail arrayOutOfBounds
+
+----------
+-- MISC --
+----------
+
+@[simp] def mem_replace_fwd (a : Type) (x : a) (_ : a) : a := x
+@[simp] def mem_replace_back (a : Type) (_ : a) (y : a) : a := y
+
+/-- Aeneas-translated function -- useful to reduce non-recursive definitions.
+ Use with `simp [ aeneas ]` -/
+register_simp_attr aeneas